Problem: If x is such that x1<2 and x1>−3, then:
Answer Choices:
A. −31<x<21
B. −21<x<3
C. x>21
D. x>21 or −31<x<0
E. x>21 or x<−31
Solution:
Method I. Let y=x1. Then (a) For y>0, when y<2,y1>21∴x>21.
(b) For y<0, when y>−3,−y<3,−y1>31,y1<−3 ∴x<−31.
Method II. Let y=x1∴xy=1; the graph is the two-branched hyperbola shown. At the point x=21,y=2. When y<2,x is to right of 21, that is, x>21. At the point x=−31,y=−3. When y>−3,x is to the left of −31, that is, x<−31.
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